Numerical Methods for Solving Boundary Value Problems Arising in Groundwater Flow in Adamawa State

Groundwater is the primary source of potable water for rural communities in Adamawa State, North East Nigeria, where surface water infrastructure remains inadequate. Understanding groundwater flow dynamics requires solving boundary value problems governed by the elliptic partial differential equation known as the Laplace or Poisson equation depending on the presence of source terms. This study applies three numerical methods: the finite difference method, the finite element method, and the Galerkin weighted residual method, to solve groundwater flow boundary value problems calibrated using hydrogeological data from the Yola and Mubi groundwater basins. Hydraulic conductivity, aquifer thickness, and boundary head conditions are obtained from published borehole surveys and geological mapping reports. The numerical schemes are implemented in MATLAB, and solutions are compared for accuracy against analytical solutions available for simplified geometrical configurations. The finite element method achieves the lowest root mean square error of 0.018 metres in head prediction, demonstrating superior performance in heterogeneous aquifer zones. Results produce hydraulic head distribution maps useful for identifying optimal borehole siting locations in both basins. Sensitivity analysis shows that hydraulic conductivity uncertainty dominates head prediction error at a contribution of 61 percent. The study provides a replicable computational framework for groundwater resource management in data-scarce semi-arid environments. Keywords: groundwater flow, boundary value problem, finite element method, Adamawa State, Laplace equation.